A student appeared in an examination consisting of 8 true-false type questions. The students guesses the answers with equal probability. The smallest value of n, so that the probability of guessing at least ‘n’ correct answers is less than $\frac{1}{2}$ is

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Detailed Solution

$P (E) < \frac{1}{2}$

$\Rightarrow \sum_{r = n}^{8} {}^{8}C_{r} {(\frac{1}{2})}^{8 - r} {(\frac{1}{2})}^{r} < \frac{1}{2} \Rightarrow \sum_{r = n}^{8} {}^{8}C_{r} {(\frac{1}{2})}^{8} < \frac{1}{2}$

$\Rightarrow {(\frac{1}{2})}^{8} ({}^{8}C_{n} + {}^{8}C_{n + 1} + ..... + {}^{8}C_{8}) < \frac{1}{2}$

$\Rightarrow {}^{8}C_{n} + {}^{8}C_{n + 1} + .... + {}^{8}C_{8} < 128$

$\Rightarrow 256 - ({}^{8}C_{0} + {}^{8}C_{1} + .... + {}^{8}C_{n - 1}) < 128$

$\Rightarrow n - 1 \geq 4 \Rightarrow n \geq 5$

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